How to get the name of image of "Image Texture" with python?
The screenshot of your texture indicates you are using cycles, the material texture slots are only used by blender internal. For a cycles material, you need to loop through the node tree and get the image info from any image texture nodes to find what images are being used.

import bpy

texture_list = []

for obj in bpy.context.scene.objects:
    for s in obj.material_slots:
        if s.material and s.material.use_nodes:
            for n in s.material.node_tree.nodes:
                if n.type == 'TEX_IMAGE':
                    texture_list += [n.image]
                    print(obj.name,'uses',n.image.name,'saved at',n.image.filepath)

print(texture_list)


If you wanted to list by face what images are used then you can loop through the objects face data -

for obj in bpy.context.scene.objects:
    if obj.type == 'MESH':
        for f in obj.data.polygons:
            mat = obj.material_slots[f.material_index].material
            if mat and mat.use_nodes:
                for n in mat.node_tree.nodes:
                    if n.type == 'TEX_IMAGE':
                        texture_list += [n.image]
                        print('Face', f.index, 'uses', n.image.name)

print(texture_list)